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Theorem infdifsn 9700
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8989 . . . 4 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
21adantr 479 . . 3 ((ω ≼ 𝐴𝐵𝐴) → ∃𝑓 𝑓:ω–1-1𝐴)
3 reldom 8980 . . . . . . 7 Rel ≼
43brrelex2i 5739 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
54ad2antrr 724 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ∈ V)
6 simplr 767 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐵𝐴)
7 f1f 6798 . . . . . . 7 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
87adantl 480 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω⟶𝐴)
9 peano1 7900 . . . . . 6 ∅ ∈ ω
10 ffvelcdm 7095 . . . . . 6 ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
118, 9, 10sylancl 584 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ 𝐴)
12 difsnen 9091 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
135, 6, 11, 12syl3anc 1368 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14 vex 3466 . . . . . . . . . 10 𝑓 ∈ V
15 f1f1orn 6854 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
1615adantl 480 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17 f1oen3g 8997 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
1814, 16, 17sylancr 585 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ ran 𝑓)
1918ensymd 9036 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ ω)
203brrelex1i 5738 . . . . . . . . . . 11 (ω ≼ 𝐴 → ω ∈ V)
2120ad2antrr 724 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ∈ V)
22 limom 7892 . . . . . . . . . . 11 Lim ω
2322limenpsi 9190 . . . . . . . . . 10 (ω ∈ V → ω ≈ (ω ∖ {∅}))
2421, 23syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ω ∖ {∅}))
2514resex 6038 . . . . . . . . . . 11 (𝑓 ↾ (ω ∖ {∅})) ∈ V
26 simpr 483 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1𝐴)
27 difss 4131 . . . . . . . . . . . 12 (ω ∖ {∅}) ⊆ ω
28 f1ores 6857 . . . . . . . . . . . 12 ((𝑓:ω–1-1𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
2926, 27, 28sylancl 584 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30 f1oen3g 8997 . . . . . . . . . . 11 (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
3125, 29, 30sylancr 585 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32 f1orn 6853 . . . . . . . . . . . . 13 (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun 𝑓))
3332simprbi 495 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto→ran 𝑓 → Fun 𝑓)
34 imadif 6643 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
3516, 33, 343syl 18 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36 f1fn 6799 . . . . . . . . . . . . . 14 (𝑓:ω–1-1𝐴𝑓 Fn ω)
3736adantl 480 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓 Fn ω)
38 fnima 6691 . . . . . . . . . . . . 13 (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
3937, 38syl 17 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ ω) = ran 𝑓)
40 fnsnfv 6981 . . . . . . . . . . . . . 14 ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4137, 9, 40sylancl 584 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4241eqcomd 2732 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
4339, 42difeq12d 4122 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4435, 43eqtrd 2766 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4531, 44breqtrd 5179 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46 entr 9037 . . . . . . . . 9 ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4724, 45, 46syl2anc 582 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48 entr 9037 . . . . . . . 8 ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4919, 47, 48syl2anc 582 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50 difexg 5334 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51 enrefg 9015 . . . . . . . 8 ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
525, 50, 513syl 18 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53 disjdif 4476 . . . . . . . 8 (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
5453a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55 difss 4131 . . . . . . . . . 10 (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56 ssrin 4235 . . . . . . . . . 10 ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
5755, 56ax-mp 5 . . . . . . . . 9 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58 sseq0 4404 . . . . . . . . 9 ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
5957, 53, 58mp2an 690 . . . . . . . 8 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
6059a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61 unen 9084 . . . . . . 7 (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
6249, 52, 54, 60, 61syl22anc 837 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
638frnd 6736 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓𝐴)
64 undif 4486 . . . . . . 7 (ran 𝑓𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
6563, 64sylib 217 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
66 uncom 4153 . . . . . . 7 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
67 eldifn 4127 . . . . . . . . . . 11 ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
68 fnfvelrn 7094 . . . . . . . . . . . 12 ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
6937, 9, 68sylancl 584 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ ran 𝑓)
7067, 69nsyl3 138 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
71 disjsn 4720 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
7270, 71sylibr 233 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
73 undif4 4471 . . . . . . . . 9 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
7472, 73syl 17 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
75 uncom 4153 . . . . . . . . . 10 ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
7675, 65eqtrid 2778 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
7776difeq1d 4120 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
7874, 77eqtrd 2766 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
7966, 78eqtrid 2778 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
8062, 65, 793brtr3d 5184 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
8180ensymd 9036 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
82 entr 9037 . . . 4 (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
8313, 81, 82syl2anc 582 . . 3 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
842, 83exlimddv 1931 . 2 ((ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
85 difsn 4807 . . . 4 𝐵𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
8685adantl 480 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
87 enrefg 9015 . . . . 5 (𝐴 ∈ V → 𝐴𝐴)
884, 87syl 17 . . . 4 (ω ≼ 𝐴𝐴𝐴)
8988adantr 479 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → 𝐴𝐴)
9086, 89eqbrtrd 5175 . 2 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
9184, 90pm2.61dan 811 1 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325  {csn 4633   class class class wbr 5153  ccnv 5681  ran crn 5683  cres 5684  cima 5685  Fun wfun 6548   Fn wfn 6549  wf 6550  1-1wf1 6551  1-1-ontowf1o 6553  cfv 6554  ωcom 7876  cen 8971  cdom 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-om 7877  df-er 8734  df-en 8975  df-dom 8976
This theorem is referenced by:  infdiffi  9701  infdju1  10232  infpss  10260
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